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Simply asking how can someone show that the vector space $V$ of all polynomials on a field, say $K$ cannot be generated with any finite set of vectors?

I don't know where to tackle the problem. :(

Thank you.

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In any finite set of polynomials, there is one of largest degree. – Jonas Meyer Dec 20 '12 at 8:14
@JonasMeyer: Ops! Yes the second one is correct. I will fix it. – Nancy Rutkowskie Dec 20 '12 at 8:16
up vote 2 down vote accepted

Suppose that it could be, for vectors (polynomials) $p_1,\ldots, p_n$. Let $$m=\text{max}\{\text{deg}(p):p=a_1p_1+\ldots+a_np_n:a_1,\ldots,a_n\in K\}.$$ Surely a polynomial of degree $m+1$ exists in $V$. This is a contradiction.

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I'd have written $m=\max\{\deg p_i\}$. – user26857 Dec 20 '12 at 8:19
@YACP I was thinking we'd still have to establish that no polynomial can exceed that degree in $V$ for the contradiction. But it is a simple proof. Same diff. – Samuel Handwich Dec 20 '12 at 8:21

Another possibility is to show that the infinite family $\{X^n : n\geq 0\}$ is linearly independent.

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